Are there logarithmic equations which can only be solved graphically?

Question

I stumbled upon this equation:

$$ log_2(x+4)=3-x $$

Naturally, I'd solve it graphically. But that got me wondering, can I also solve it algebraically and I honestly couldn't.

I tried rewriting it, so I could "see" where to go, but I didn't get anywhere.

$$ \frac{ln(x+4)}{ln(2)} = e^{ln(3-x)} $$

$$ 2^{3-x} = x+4 $$

Now, looking at similar posts, I noticed that everybody said, it is much easier to just graph it, but a question emerges:

Are there equations, which can not be solved algebraically, but only graphically?

and if it is possible to solve this algebraically, please let me know how to do it

Answer 1

Hint: One can solve this using the so called Lambert W-function we have $$x=\frac{W(128 \log (2))-4 \log (2)}{\log (2)}$$

Answer 2

Sure, plenty of things we can't solve algebraically, which is why we do our best to approximate using approximation methods.

One such example is trying to solve $x^x=\Gamma(x)$, which can't be done with anything like the Lambert W function.

As a side note, what I would consider to be one of the simplest algebraic methods to finding solutions is fixed point iteration. For a simple example,

$$x^2=5$$

$$2x^2=5+x^2$$

$$x=\frac{5+x^2}{2x}$$

Suppose you think the solution is around $x=2$ (from graphing or guesswork), then see that

$$x\approx\frac{5+2^2}{2\times2}=\frac94$$

But if $x\approx\frac94$, then...

$$x\approx\frac{5+\left(\frac94\right)^2}{2\times\frac94}=\frac{161}{72}$$

etc.

$$x\approx\frac{5+\left(\frac{161}{72}\right)^2}{2\times\frac{161}{72}}=2.23606797792$$

The correct solution to $x^2=5$ is $x=\sqrt5=2.2360679775$.

And of course, there are other methods to doing this, but some require more than algebra.